Sy-David Friedman

If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$, then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$, for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$. This is accomplished through a use of David's trick and a coding through the ∑ 2 stable ordinals of L.

This paper investigates when it is possible for a partial ordering P to force Pκ(λ) \ V to be stationary in VP. It follows from a result of Gitik that whenever P adds a new real, then Pκ(λ) \ V is stationary in VP for each regular uncountable cardinal κ in VP and all cardinals λ &gt κ in VP [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω₁-Erdős cardinals: If P is N₁-Cohen forcing, then Pκ(λ) \ V is stationary in VP, for all regular κ ≥ N₂ and all λ &gt κ. The following is equiconsistent with an ω₁-Erdős cardinal: If P is N₁-Cohen forcing, then PN₂ (N₃) \ V is stationary in VP. The following is equiconsistent with κ measurable cardinals: If P is κ-Cohen forcing, then Pκ + (Nκ \ V is stationary in VP

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on , the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on . In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is that of the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal α the existence of singletons whose α-jumps are Turing incomparable

By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model.

Vopenka [2] proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over, and indeed over the even smaller inner model $\mathbb{S}=$, where S is the Stability predicate. We refer to the inner model $\mathbb{S}$ as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core.

The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals

In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:Vλ+1≺Vλ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j : V_{\lambda+1} {\prec} V_{\lambda+1}}$$\end{document} with the failure of GCH at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}.